Simplify Square Root Of 42

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Simplifying a foursquare root isn't equally hard every bit it looks. To simplify a square root, yous merely have to factor the number and pull the roots of whatever perfect squares you find out of the radical sign. Once you've memorized a few mutual perfect squares and know how to factor a number, you'll be well on your way to simplifying the square root.

1

Understand factoring. The goal of simplifying a square root is to rewrite it in a form that is piece of cake to sympathise and to use in math problems. Factoring breaks down a big number into 2 or more smaller factors, for instance turning 9 into 3 x 3. One time we find these factors, nosotros tin rewrite the square root in simpler course, sometimes fifty-fifty turning information technology into a normal integer. For example, √9 = √(3x3) = 3. Follow the steps beneath to learn this procedure for more than complicated foursquare roots.^[one]
ii
Divide by the smallest prime number possible. If the number under the foursquare root is even, dissever it by two. If your number is odd, try dividing it past 3 instead. If neither of these gives you a whole number, movement down this listing, testing the other primes until you get a whole number consequence. You only need to exam the prime numbers, since all other numbers have prime number numbers as their factors. For example, you don't need to examination 4, because any number divisible by iv is also divisible by two, which you already tried.^[two]
- 2
- iii
- 5
- seven
- 11
- 13
- 17
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3

Rewrite the square root as a multiplication problem. Keep everything underneath the square root sign, and don't forget to include both factors. For example, if y'all're trying to simplify √98, follow the step above to find that 98 ÷ 2 = 49, so 98 = two x 49. Rewrite the "98" in the original square root using this data: √98 = √(2 x 49).^[iii]
4
Repeat with one of the remaining numbers. Before we can simplify the square root, we keep factoring information technology until we've broken it down into two identical parts. This makes sense if you call up virtually what a square root means: the term √(ii x ii) means "the number you can multiply with itself to equal 2 10 2." Plain, this number is 2! With this goal in mind, allow's repeat the steps above for our example trouble, √(2 ten 49):
- two is already factored as low every bit it will go. (In other words, it's one of those prime number numbers on the list above.) We'll ignore this for now and effort to divide 49 instead.
- 49 can't be evenly divided past 2, or by iii, or by 5. Y'all can test this yourself using a calculator or long sectionalisation. Considering these don't give us overnice, whole number results, nosotros'll ignore them and keep trying.
- 49 tin can be evenly divided by 7. 49 ÷ 7 = 7, and so 49 = 7 ten 7.
- Rewrite the trouble: √(two ten 49) = √(2 10 7 ten 7).
5
Finish simplifying past "pulling out" an integer. In one case you've broken the problem down into two identical factors, yous tin can turn that into a regular integer outside the foursquare root. Get out all other factors within the foursquare root. For example, √(2 x 7 ten 7) = √(ii)√(7 ten vii) = √(ii) x 7 = 7√(2).^[four]
- Fifty-fifty if it's possible to keep factoring, you lot don't need to in one case you've found ii identical factors. For example, √(16) = √(four 10 iv) = 4. If we kept on factoring, we'd finish up with the same answer but accept to do more work: √(sixteen) = √(4 ten 4) = √(2 10 2 x 2 x two) = √(two x 2)√(2 x 2) = ii ten 2 = 4.
six
Multiply integers together if at that place are more than one. With some large foursquare roots, you can simplify more than one time. If this happens, multiply the integers together to get your final problem. Hither's an example:
- √180 = √(2 x 90)
- √180 = √(2 x 2 x 45)
- √180 = 2√45, but this can withal be simplified farther.
- √180 = 2√(iii x 15)
- √180 = 2√(three ten 3 10 5)
- √180 = (2)(3√5)
- √180 = 6√5
vii
Write "cannot be simplified" if at that place are no ii identical factors. Some square roots are already in simplest form. If yous keep factoring until every term under the foursquare root is a prime (listed in ane of the steps above), and no two are the same, then there's zippo yous can do. You might have been given a trick question! For example, permit's try to simplify √lxx:^[5]
- 70 = 35 x 2, and so √lxx = √(35 x 2)
- 35 = seven x 5, then √(35 x two) = √(7 x 5 x ii)
- All 3 of these numbers are prime number, and then they cannot be factored farther. They're all different, then at that place's no way to "pull out" an integer. √70 cannot be simplified.

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1
Memorize a few perfect squares. Squaring a number, or multiplying it by itself, creates a perfect foursquare. For example, 25 is a perfect square because 5 x 5, or five^two, equals 25. Memorizing at least the first 10 perfect squares can aid you recognize and quickly simplify perfect square roots. Here are the starting time ten perfect squares:
- one^two = 1
- 2^two = 4
- iii² = 9
- 4² = 16
- 5ⁱⁱ = 25
- six² = 36
- 7ⁱⁱ = 49
- eight² = 64
- ix² = 81
- x² = 100
2
Find the square root of a perfect square. If y'all recognize a perfect square under a square root symbol, y'all can immediately plough it into its square root and get rid of the radical sign (√). For case, if you see the number 25 under the square root sign, y'all know that the reply is v because 25 is a perfect square. Here'southward the same list every bit above, going from the square root to the respond:^[6]
- √one = 1
- √4 = two
- √ix = 3
- √16 = iv
- √25 = 5
- √36 = 6
- √49 = 7
- √64 = 8
- √81 = 9
- √100 = 10
3
Factor numbers into perfect squares. Apply the perfect squares to your reward when following the factor method of simplifying square roots. If you lot notice a style to factor out a perfect foursquare, information technology can save you time and effort. Hither are some tips:^[7]
- √fifty = √(25 x 2) = v√2. If the concluding 2 digits of a number end in 25, l, or 75, yous tin ever gene out 25.
- √1700 = √(100 x 17) = 10√17. If the final two digits finish in 00, y'all can e'er factor out 100.
- √72 = √(9 x 8) = 3√eight. Recognizing multiples of nine is often helpful. There's a pull a fast one on to it: if all digits in a number add upward to nine, then ix is always a factor.
- √12 = √(four ten three) = ii√3. There's no special play a trick on here, just information technology'due south normally easy to check whether a small number is divisible by four. Keep this in mind when looking for factors.
4
Factor a number with more than one perfect square. If the number's factors contain more one perfect square, motion them all exterior the radical symbol. If y'all plant multiple perfect squares during your simplification process, motility all of their square roots to the outside of the √ symbol and multiply them together. For example, let's simplify √72:
- √72 = √(9 x 8)
- √72 = √(nine x iv 10 2)
- √72 = √(9) x √(4) x √(2)
- √72 = iii ten ii x √two
- √72 = 6√ii

1

Know that the radical symbol (√) is the square root symbol. For example, in the trouble, √25, "√" is the radical symbol.^[8]
2

Know that the radicand is the number inside the radical symbol. You lot will need to find the foursquare root of this number. For case, in the problem √25, "25" is the radicand.^[9]
iii

Know that the coefficient is the number outside the radical symbol. This is the number that the square root is being multiplied by; this sits to the left of the √ symbol. For case, in the problem, 7√ii, "7" is the coefficient.
iv

Know that a gene is a number that can exist evenly divided out of some other number. For example, ii is a gene of eight because eight ÷ 4 = 2, but iii is not a cistron of 8 because viii÷three doesn't upshot in a whole number. As another example, 5 is a gene of 25 because 5 x five = 25.
5

Sympathize the meaning of simplifying a square root. Simplifying a square root just ways factoring out any perfect squares from the radicand, moving them to the left of the radical symbol, and leaving the other gene inside the radical symbol. If the number is a perfect square, then the radical sign will disappear once you write downwardly its root. For case, √98 can be simplified to seven√2.

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Question

How do you simplify radical fractions?

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Offset past multiplying the numerator and denominator by a radical that will eliminate the radical in the denominator. For example, if your fraction is 2/√vii, multiply by √vii/√7 to get 2√7/√49. This will simplify to 2√7/7. If possible, simplify the fraction by dividing out whatever mutual factors in the numerator and denominator.
Question

What is the simplest radical grade?

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Staff Answer

If something is written in its simplest radical course, that means that y'all have already constitute all possible roots and eliminated any radicals from the denominator of a fraction. A square root can't be simplified whatsoever further if in that location are no ii identical factors remaining and every term nether the radical symbol is a prime number.
Question

How do yous simplify square roots with variables?

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Staff Reply

Apply the aforementioned procedure you lot would for simplifying square roots with numerical radicands. Bring any identical pairs of factors out of the radical to become the coefficient. For example, if you need to simplify √25a^3, change it to √5×v×a×a×a. Cistron out five and a^two to get 5a^2√a.

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One fashion to detect perfect squares that factor into a number is to expect through the list of perfect squares, beginning with the 1 that is the next smallest compared to your radicand, or the number nether the square root sign. For example, when looking for the perfect square that goes into 27, you might commencement at 25 and go down the list to xvi and stop at 9, when you found the one that divides into 27.

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Calculators can be useful for big numbers, just the more you practice working this out on your own, the easier this will become.

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Simplifying is not the aforementioned as evaluating. At no point in this process should y'all go a number with a decimal point in it!

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Article Summary X

To simplify a square root, start by dividing the square root by the smallest prime number number possible. For example, if you're trying to find the foursquare root of 98, the smallest prime possible is 2. If you divide 98 past two, you go 49. Then, rewrite the square root as a multiplication problem nether the square root sign. In this case, yous'd rewrite the square root as two × 49 under the foursquare root sign. From there, go on factoring the numbers until you have 2 identical factors. In this example, 49 divided by 7 is vii. Rewrite the square root every bit 2 × 7 × vii. Finally, once you lot accept two identical numbers, move them outside of the foursquare root to make them a regular integer. So the simplified square root of 98 is seven × the square root of two. Notwithstanding, if you factor the numbers within the square root as much as you can without getting two identical numbers, then your square root can't be simplified! To larn other ways you can simplify a square root, keep reading!

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